qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,016,588 | <p>I'd really appreciate a push in the right direction for solving this. I just can't get it. Thanks</p>
<p>Prove $ ({x+1})^{1/3} < 1 + {\frac13}x $ for x > 0 </p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$1-\sin2x=1-\cos2\left(\dfrac\pi4-x\right)=2\sin^2\left(\dfrac\pi4-x\right)=\dfrac2{\csc^2\left(\dfrac\pi4-x\right)}$$</p>
<p>As $\csc(-A)=-\csc A,$</p>
<p>$$\csc^2\left(\dfrac\pi4-x\right)=\csc^2\left(x-\dfrac\pi4\right)$$</p>
<p>$$\int\csc^2y\ dy=-\cot y+K$$</p>
<hr>
<p>Alternatively,</p>
<p>$... |
2,016,588 | <p>I'd really appreciate a push in the right direction for solving this. I just can't get it. Thanks</p>
<p>Prove $ ({x+1})^{1/3} < 1 + {\frac13}x $ for x > 0 </p>
| Community | -1 | <p>$$\int\frac{dx}{(\cos x-\sin x)^2}=\int\frac{dx}{(\sqrt2\cos(x+\frac\pi4))^2}$$ where you recognize the derivative of a tangent.</p>
<hr>
<p>A very general and useful method is the use of the exponential representation</p>
<p>$$\cos x:=\frac{e^{ix}+e^{-ix}}2,\\\sin x:=\frac{e^{ix}-e^{-ix}}{2i}$$ together with the... |
3,881,711 | <p>Sorry in advance, it is probably a stupid question.
I encountered it when I was thinking about the birthday problem. The probability of having at least one pair of the same birthday is
<span class="math-container">$$ 1- \frac{365\cdot364\cdot\ldots\cdot(365-n+1)}{365^n}$$</span> and it is above 0.5 for n>22.
Howe... | Philipp | 579,544 | <p>You can apply the direct comparision test if you construct a suitable series which forms an upper bound.</p>
<p>Let be <span class="math-container">$n>1$</span>, then by definition we know that <span class="math-container">$e^{x}:=\sum\limits_{j=0}^{\infty}\frac{x^j}{j!}=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2... |
2,182 | <p>I wondered if it is appropriate to ask on mathematica stack exchange a question about what they think about the ergonomy of mathematica in comparison to other softwars (matlab etc).</p>
<p>Because I find mathematica very unfriendly in comparison of everything I learnt but I would have to have other point of view to... | rcollyer | 52 | <p><strong>Full merge option</strong>: </p>
<ul>
<li>Make <a href="https://mathematica.stackexchange.com/questions/tagged/geographics" class="post-tag" title="show questions tagged 'geographics'" rel="tag">geographics</a> a synonym to <a href="https://mathematica.stackexchange.com/questions/tagged/graphics" cl... |
2,117,784 | <p>A <strong>multilinear form</strong> is a mapping</p>
<p>\begin{align}
\Delta: V^n \rightarrow K
\end{align}</p>
<p>where $V$ is a finite-dimensional vector space over field $K$.</p>
<p>It must meet the following requirements:</p>
<ul>
<li>First:</li>
</ul>
<p>\begin{align}
&\Delta\left(a_1, \dots, a_{i-1}, ... | DIEGO R. | 297,483 | <p>Let's denote the vector space <span class="math-container">$W$</span> by <span class="math-container">$L^{n}(V_{m}, \mathbb{K})$</span>, where <span class="math-container">$m = \dim V$</span>. We claim that <span class="math-container">$\dim L^{n}(V_{m}, \mathbb{K}) = m^{n}$</span>. Let <span class="math-container">... |
3,767,129 | <p>Problem:</p>
<p>The vertex of a pyramid lies at the origin, and the base is perpendicular to the x-axis at <span class="math-container">$x = 4$</span>. The cross sections of the pyramid perpendicular to the x-axis are squares whose diagonals run from the curve <span class="math-container">$y = -5x^2$</span> to the c... | Anatoly | 90,997 | <p>Your method is correct but you lost a <span class="math-container">$1/2$</span> factor. The value of <span class="math-container">$10x^2$</span> is the diagonal, not the side of the square. So the integrand becomes <span class="math-container">$\frac 12 (10x^2)^2$</span>.</p>
|
628,447 | <p>I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula</p>
<p>$$H= I - (uu^T/\beta),$$</p>
<p>they are not equal. What's wrong with my reasoning?</p>
<p>EDIT: I forgot that $(uu^T)^T$ would be $(u^T)^T(u)^T$ from the following properties:
$(AB)^T=B^TA^T$</p>
| Listing | 3,123 | <p>$$H = I - \frac {2} {u^T u} u u^T=I+\alpha \ u u^T$$</p>
<p>$$H^T=(I+\alpha \ u u^T)^T=I^T+\alpha \ (uu^T)^T=I+\alpha (u^T)^Tu^T=I+\alpha \ u u^T=H$$</p>
<p>Where I used $I^T=I$ and the <a href="http://en.wikipedia.org/wiki/Transpose#Properties" rel="nofollow">basic properties</a> of the transposed matrix, namely:... |
63,542 | <p>I'm trying to solve for the output of a simple RC low-pass filter, with Cos[w t] input. The convolution of the input with the RC transfer function gives me the right output, but I'd like to be able to automate getting it in the right equivalent form. Specifically, a phase-shifted Cosine function.</p>
<p>The convolu... | Mr.Wizard | 121 | <p>Pattern matching takes place on (something close to) the <a href="http://reference.wolfram.com/language/ref/FullForm.html" rel="nofollow noreferrer"><code>FullForm</code></a> of the expression rather than the display form that you see. You can visualize it using <a href="http://reference.wolfram.com/language/ref/Tr... |
63,542 | <p>I'm trying to solve for the output of a simple RC low-pass filter, with Cos[w t] input. The convolution of the input with the RC transfer function gives me the right output, but I'd like to be able to automate getting it in the right equivalent form. Specifically, a phase-shifted Cosine function.</p>
<p>The convolu... | kglr | 125 | <p>Changing <code>a__</code> and <code>b__</code> to <code>a_</code> and <code>b_</code>, respectively,</p>
<pre><code>j2[q_] := q /. b_ Cos[x__] + a_ Sin[x__] :> {{a}, {b}, {x}};
j2[h]
</code></pre>
<p>gives</p>
<blockquote>
<p>$\left( \frac{\tau w}{\tau ^2 w^2+1}, \frac{1}{\tau ^2 w^2+1}, t w \right) $</p>
<... |
1,625,306 | <p>I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$:</p>
<p>$$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$
I attempted to change the function to $[\sin(ax+x)+\sin(ax-x)]/2$ and then integrate, but I am left with (-)cosines with a zero in the denominato... | TZakrevskiy | 77,314 | <p>You can find the indefinite integral via integration by parts:</p>
<p>$$\int \cos x \sin (ax)dx = \sin x \sin (ax) -\int a \sin x \cos (ax)dx$$</p>
<p>$$= \sin x \sin (ax) - a\left(-\cos x \cos(ax)-a\int\cos x \sin(ax)dx \right),$$
which leads to </p>
<p>$$\int \cos x \sin (ax)dx = a^2 \int \cos x \sin (ax)dx +\s... |
751,063 | <p>Let $G$ be a finite dimensional connected Lie group and $Diffeo(G)$ be the diffeomorphism group of the underlying manifold. Is it true that $Diffeo(G)$ has the homotopy type of a finite dimensional Lie group? I can't seem to find a counterexample.</p>
| Moishe Kohan | 84,907 | <p>Torus of dimension $\ge 25$ is a counter example. See <a href="http://www.ams.org/journals/bull/1970-76-06/S0002-9904-1970-12621-0/S0002-9904-1970-12621-0.pdf" rel="nofollow">here</a>.</p>
|
4,634,293 | <p>Given a set <span class="math-container">$A$</span> of cardinality <span class="math-container">$n$</span>, let <span class="math-container">$\mathbb{P}(A)$</span> be the power set of <span class="math-container">$A$</span>. What is the number of edges of the intersection graph of the powerset of <span class="math-c... | Mees de Vries | 75,429 | <p>Although not as conceptually clear as the other answer, another approach is to use induction. Suppose that the count is correct for <span class="math-container">$A = \{1,\ldots, n-1\}$</span> and consider the graph corresponding to <span class="math-container">$A \cup \{n\}$</span>.</p>
<p>First, if <span class="mat... |
610,029 | <p>The time between successive cars on a certain road is exponentially distributed and the probability is $1/2$ that the next car will arrive within two minutes. Assume the time between and particular pair of cars is independent of the times between all other pairs of cars. </p>
<ol>
<li><p>What is the probability the... | Community | -1 | <p>First, note that we have $a^2=c^2-b^2=6b+9$. We want this to be a perfect square; it's easy to see that we must have $3|b$ and modular considerations show we have to have $2|n$; a little computation suggests to us that $b=6n(n+1)$ is always a perfect square, with $a^2=36n^2+36n+9=9(2n+1)^2$. Now $a^2+b^2=36n^2+36n+9... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Kevin | 198,741 | <blockquote>
<p>And if so, is there any guide or principle that should tell us when to use words or symbols in our proofs?</p>
</blockquote>
<p>It greatly depends on what you are trying to do with your proofs.</p>
<p>If you "just want to do math," then you should use whatever combination of symbols and words is cle... |
1,046,229 | <p>For this problem in proving that the cardinality of <span class="math-container">$(0,1)$</span> is equal to that of the set of real numbers, would I just prove that <span class="math-container">$(0,1)$</span> is uncountable, and then use the theorem that the subset of an uncountable set is uncountable, by saying <sp... | Daniel McLaury | 3,296 | <p>No, because "uncountable" just means "bigger than countable." There are lots of different uncountable cardinalities.</p>
<p>You need to construct an explicit bijection between (0, 1) and $\mathbb{R}$, or I guess you could construct injective maps in both directions and apply Cantor–Schroeder–Bernstein.</p>
|
1,864,321 | <p>I tried to prove the following inequality which gives a lower bound to the Mathieu sum:
$$S=\sum_{k=1}^\infty\dfrac{2k}{(k^2+c^2)^2}$$
where $c\neq0$.
The Mathieu inequality states: $S\lt\dfrac{1}{c^2}$
The following inequality holds:
$$S\gt\dfrac{1}{c^2+\dfrac{1}{2}}$$
I tried to expand $S$ and I found an expressi... | Marco Cantarini | 171,547 | <p>You may observe that $$\frac{2k}{\left(k^{2}+c^{2}\right)^{2}+\frac{1}{4}+c^{2}}<\frac{2k}{\left(k^{2}+c^{2}\right)^{2}}\tag{1}
$$ and $$\frac{2k}{\left(k^{2}+c^{2}\right)^{2}+\frac{1}{4}+c^{2}}=\frac{1}{\left(k-\frac{1}{2}\right)^{2}+\frac{1}{4}+c^{2}}-\frac{1}{\left(k+\frac{1}{2}\right)^{2}+\frac{1}{4}+c^{2}}
... |
4,095,486 | <p>I am currently learning about Jacobians, and I need help on the following integral:</p>
<blockquote>
<p><span class="math-container">$$
\int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy.
$$</span></p>
</blockquote>
| Richard | 897,911 | <p>There is no way to represent <span class="math-container">$\int\cos(x^2)dx$</span> in terms of elementary functions! This is called the Fresnel integral, see <a href="https://en.wikipedia.org/wiki/Fresnel_integral" rel="nofollow noreferrer">Fresnel Integral</a>. The best you can do is use series expansions.</p>
|
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | Anant Gupta | 683,800 | <p>Let us take an example of a vector V = i + 2j which in matrix notation is </p>
<p><a href="https://i.stack.imgur.com/RTUxD.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTUxD.gif" alt="enter image description here"></a></p>
<p>It will be represented as
<a href="https://i.stack.imgur.com/BoCir.... |
3,318,888 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow \mathbb{R}$</span> be such that</p>
<p>(1) <span class="math-container">$f$</span> is bounded.</p>
<p>(2) <span class="math-container">$f$</span> is integrable on <span class="math-container">$[\delta,1]$</span> for every <span class="math-container"... | Brian Moehring | 694,754 | <p>Write <span class="math-container">$\overline{\int_a^b} f(x)\,dx$</span> and <span class="math-container">$\underline{\int_a^b} f(x)\,dx$</span> for the upper and lower Riemann integrals, respectively. Since <span class="math-container">$f$</span> is bounded, there is some <span class="math-container">$M$</span> su... |
2,943,037 | <p>Statement : Prove that <span class="math-container">$SL(n,\mathbb{Z})$</span> is generated by <span class="math-container">$(n^2-n)$</span> elements.</p>
<p>The determinant is a n linear function of the rows of the matrix. Given any matrix, if the determinant is nonzero, say <span class="math-container">$det(x_1, x... | Angina Seng | 436,618 | <p>If <span class="math-container">$A$</span> is a real symmetric matrix, then <span class="math-container">$\exp(A)$</span> is positive definite.
For <span class="math-container">$A=UDU^{-1}$</span> where <span class="math-container">$D$</span> is diagonal and <span class="math-container">$U$</span> is orthogonal, and... |
2,746,388 | <p>A common tangent to two curves is a line that is tangent to the two curves, but not necessarily at the same point.</p>
<p>Find, in terms of $a$ and $b$, the explicit equation of the common tangent to the two curves $y = x^2 + ax + b$ and $y = x^2 + bx + a$, where $a$ is not equal to $b$.</p>
<p>Also find, in terms ... | amd | 265,466 | <p>Before grinding through some generic calculation, it’s a good idea to look for special features of the problem that will simplify it. In this case, the two curves are parabolas that are related by a translation. Any common tangent line must be invariant under this translation, which implies that it must be parallel ... |
2,675,565 | <p>I have a specific problem and I am kind of stuck. Don't know exactly where to begin defining what it is. Is someone could just give me a nodge in the right direction or even better, tell me what kind of problem it is, that would be nice. I can of course do the hard work myself but right now I need some help figuring... | user2825632 | 250,232 | <p>First, find every possible combination of sets that satisfy the requirement for number of red balls. For your example, these are:</p>
<p>$S_1 = \{1, 2\}$</p>
<p>$S_2 = \{1, 4\}$</p>
<p>$S_3 = \{1, 5\}$</p>
<p>$S_4 = \{2, 4, 5\}$</p>
<p>$S_5 = \{1, 2, 3\}$</p>
<p>$S_6 = \{1, 3, 4\}$</p>
<p>$S_7 = \{1, 4, 5\}$<... |
2,145,549 | <p>I'd like to ask a question about what can I possibly do wrong with determining asymptotes of the function</p>
<p>$$x \mapsto x-2\sqrt{x^2+1} $$</p>
<p>OK, so when it comes to vertical asymptotes function, we can't have any because domain of the function is the set of all real numbers.</p>
<p>Now, I'm trying to de... | Travis Willse | 155,629 | <p>What is wrong is that the computation is that, contrary to what is claimed,
$$\frac{\left(x-2\sqrt{x^2+1}\right)\left(x-2\sqrt{x^2+1}\right)} {\left(x-2\sqrt{x^2+1}\right)} \neq \frac {-3x^2-4}{3x+2}.$$
For one, when manipulating this way, one wants to multiply the numerator and denominator by the <em>conjugate</em>... |
2,981,554 | <p>This might be silly, but I am not sure:</p>
<p>Does there exist a Lebesgue measurable subset <span class="math-container">$E \subseteq (0,1)$</span> such that</p>
<ol>
<li><p><span class="math-container">$E$</span> and <span class="math-container">$(0,1) \setminus E$</span> both have positive Lebesgue measure.</p>... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$E$</span> be the union of <span class="math-container">$(0,1) \setminus \mathbb Q)\cap (0,\frac 1 2]$</span> and <span class="math-container">$\mathbb Q\cap (\frac 1 2,1)$</span>. Then <span class="math-container">$E$</span> and <span class="math-container">$(0,1)\setminus E$</sp... |
22,892 | <p>Imagine the following scenario:
user A is developing a project with documentation in workbench. She wants to give her (unfinished) work to user B , so that user B can continue work on the project and eventually deploy it on her PC. The 2 users might have different operating systems.
What is the best way to accompli... | Verbeia | 8 | <p>Import/Export is the development equivalent of sneaker-net. While exporting and importing the projects works fine, you might want to consider installing some form of source control into your Workbench/Eclipse environment. You then do not need to explicitly "export" anything, just check-in your changes as you save.</... |
279,573 | <p>I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:</p>
<ol>
<li><p>$X$ has the same Betti numbers as $\mathbb{C}\mathbb{P}^{3}$ i.e. $b_{1}(X) = b_{3}(X) = 0$ and $b_{2}(X) = 1$ and all of its cohomology groups are tors... | Nick L | 99,732 | <p>As pointed out by dhy, the question is completely resolved in section 3 of "Uniformization of Fake Projective Four Spaces", by Sai-Kee Yeung. The conclusion is that there is no such $3$-fold.</p>
|
3,123,120 | <p>Prove that given sequence <span class="math-container">$$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$</span> </p>
<p>is a Cauchy sequence </p>
<p>My attempt :
<span class="math-container">$|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\dfrac{(-1)^{m+1}}{m+2}\cdots\dots+\dfra... | Minus One-Twelfth | 643,882 | <p>We have <span class="math-container">$f(w) = \sigma(x^T w)$</span> (remember that <span class="math-container">$w^Tx= x^T w)$</span>. Hence the gradient vector with respect to <span class="math-container">$w$</span> is
<span class="math-container">$$
\begin{align*}
\frac{\partial}{\partial w}f(w) &= \sigma'(x^T ... |
3,123,120 | <p>Prove that given sequence <span class="math-container">$$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$</span> </p>
<p>is a Cauchy sequence </p>
<p>My attempt :
<span class="math-container">$|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\dfrac{(-1)^{m+1}}{m+2}\cdots\dots+\dfra... | Robert Lewis | 67,071 | <p>With</p>
<p><span class="math-container">$x = (x_1, x_2, \ldots, x_n)^T \tag 1$</span></p>
<p>and</p>
<p><span class="math-container">$w = (w_1, w_2, \ldots, w_n)^T, \tag 2$</span></p>
<p>we have</p>
<p><span class="math-container">$w^Tx = \displaystyle \sum_1^n w_i x_i; \tag 3$</span></p>
<p>we observe that</... |
4,397,783 | <p>A car salesman can make a sale to 65% of his male customers but to only 45% of his female customers. All of his sales are independent. On Monday morning, the car salesman has two make and one female customer. Find the probability that he makes exactly two sales.</p>
<p>My Solution:
Lets Call the two male customers T... | awkward | 76,172 | <p>Hint: Notice that
<span class="math-container">$$(.35 + .65 x)^2 (.55 + .45 x) = 0.067375\, +0.305375 x+0.437125 x^2+0.190125 x^3$$</span></p>
<p>The coefficient of <span class="math-container">$x^2$</span> matches your book's answer. Maybe this is not a coincidence.</p>
|
2,684,433 | <p>If I have a set $\{(1, H), (2, C), (3, F), (4, Z), (5, S), (6, L) \}$ is there any way to express this with set builder notation? If not, is there any other way to express this mathematically?</p>
| B. Goddard | 362,009 | <p>If you let $m=n(n-1)$ then your polynomial is $m(m+1)(m+2)$. This is a multiple of $3$, which you could show by induction. And $m$ is even, therefore $m+2$ is even and one of them must be a multiple of $4$, so the expression is a multiple of $8$. It's probably easier to show both of these facts by induction than ... |
4,627,884 | <p>Consider the 'pseudo' definitions below:</p>
<p><span class="math-container">$A$</span> is an <span class="math-container">$n \times n$</span> matrix</p>
<p><span class="math-container">$A$</span> is <span class="math-container">$m_1$</span> if <span class="math-container">$A^2= I$</span> (the identity matrix)</p>
<... | Michael Stachowsky | 337,044 | <p>Yes, there are many such matrices. These are known as "self-inverse" matrices. I can provide an example:</p>
<p>A rotation matrix <span class="math-container">$R \in \mathbb{R}^{2x2}$</span> rotates vectors by an angle <span class="math-container">$\theta$</span>. The rotation matrix that rotates vectors b... |
691,497 | <p>Suppose that $M$ and $F$ are real matrices. Let $A$ be the block-matrix
$$
A=
\begin{pmatrix}
M & F \\
F & M
\end{pmatrix}
$$
If $\det(M)=0$ is $\det(A)\leq0$? If not, what conditions need to be satisfied?</p>
<p>Also, </p>
<p>Does A have a non-positive eigenvalue?</p>
| Brian Fitzpatrick | 56,960 | <p>Let
$$
M=
\begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}\quad
F
=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
$$
Then
$$
\det
\begin{pmatrix}
M & F \\
F & M
\end{pmatrix}=1
$$</p>
|
2,759,636 | <p>I attempted to use Pascal's triangle identity to help out, but I do not know how to deal with $\frac{1}{n+1}$.</p>
| Misha Lavrov | 383,078 | <p>The combinatorial approach is to notice that $\frac1{n+1} \binom{2n}{n}$ is the $n^{\text{th}}$ <a href="https://en.wikipedia.org/wiki/Catalan_number" rel="nofollow noreferrer">Catalan number</a>, so it's an integer because it counts something.</p>
<p>Algebraically, one possible approach is to notice that
$$
\fr... |
3,502,232 | <p>I've tried to apply Bayes theorem to the following question, but I think I'm using it wrong.</p>
<p>2000 people take an exam and 1 person is cheating. A lie detector that is accurate 99% of the time is used to screen the candidates one by one. At some point during the screening the lie detector beeps to signal that... | Doug M | 317,176 | <p>Until you get to be very comfortable applying Bayes theorem, I suggest you fill out this table.</p>
<p><span class="math-container">$\begin {array}{cc|c}
\text {True Positive} & \text {False Positive} & \text {Measured Positives}\\
\text {False Negative} & \text {True Negative} & \text {Measured Neg... |
3,798,735 | <p>Let <span class="math-container">$G(x,y): \mathbb{R}_0^+ \times \mathbb{R}_0^+ \to \{0,1,\infty\}$</span> be a (Borel) measurable function. If the set of values <span class="math-container">$(x,y) \in \mathbb{R}_0^+ \times \mathbb{R}_0^+ $</span> such that <span class="math-container">$G(x,y) = \infty$</span> has po... | Christoph | 86,801 | <p>Indeed, after defining an object or notion by certain properties you can then make the true statement that the defined object or notion has these properties. The statement is then said to be "true by definition".</p>
<p>However, you asked "Can definitions become statements?", to which the answer ... |
4,129,851 | <p>This was a problem in my textbook.</p>
<p>Suppose we had a bag with <span class="math-container">$2$</span> balls, an orange and a blue ball. If we pick a blue ball, we simply put it back. If we select an orange ball, we put it back but add <strong>another</strong> orange ball. Suppose we do this <span class="math-c... | Meowdog | 803,787 | <p>You integrate by parts:
<span class="math-container">$$
\int^1_0 e^{x^2} \sin(nx)~\mathrm{d}x = - \frac{1}{n}(e \cos(n) -1) + \frac{1}{n} \int^1_0 2x e^{x^2}\cos(nx)~\mathrm{d}x
$$</span>
Therefore, by using the triangle inequality, <span class="math-container">$\lvert \cos(nx) \rvert \leq 1$</span> and <span class=... |
4,666 | <p>It looks to me like a number in a base other than base 10 gets evaluated before the evaluator ever gets a chance to be tweaked.</p>
<p>For example, <code>FullForm[16^^abcdef]</code> or even <code>FullForm[HoldAll[16^^abcdef]]</code> both produce <code>11259375</code>.</p>
<p>Am I missing a trick that would get me ... | Szabolcs | 12 | <p>To understand what's happening, the difference between evaluation and parsing needs to be made clear:</p>
<ul>
<li><p><a href="http://en.wikipedia.org/wiki/Parsing"><em>parsing</em></a> means taking the string (the text) input to Mathematica and converting it to some internal representation of a Mathematica express... |
275,473 | <p>Let $f$ and $g$ be two functions with derivatives in some interval containing $0$, where $g$ is positive. Also</p>
<p>$$f(x)=o(g(x))~as~x \rightarrow0$$</p>
<p>Prove or dissprove:</p>
<p>1) $$\int_0^xf(t)dt=o\left(\int_0^xg(t)dt\right)$$
2) $$f'(x)=o(g'(x))$$</p>
<p>Now considering the first, my reasoning is as ... | Sarunas | 49,864 | <p>Considering the second - I can not use the L'Hopital's rule yet, so I have to stick to other means. I think I found away, but I need help confirming weather such approach is viable:</p>
<p>Let $F(x)=\int_0^xf(t)dt$ and $G(x)=\int_0^xg(t)dt$. By definition:</p>
<p>$$\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}=F'(x)+o(1)$$
$... |
429,138 | <p>I need help with the steps of finding the slope and $y$-intercept of this equation: $f(x)=3x-\frac{1}{5}$. I am in Algebra two and do not understand the proper ways, or steps of doing this.</p>
| Clayton | 43,239 | <p><strong>HINT:</strong> Do you recognize the equation $y=mx+b$? If so, what are the $m$ and $b$?</p>
|
429,138 | <p>I need help with the steps of finding the slope and $y$-intercept of this equation: $f(x)=3x-\frac{1}{5}$. I am in Algebra two and do not understand the proper ways, or steps of doing this.</p>
| john | 79,781 | <p>In the equation $f(x)=m x+c$ the slope is given by $m$ and the y-intercept by $c$. This is because the slope is how much the function increases by each time we increase $x$ by one (try this for any two values of $x$ that are one apart) and the y-intercept is the value of $f(x)$ when $x=0$.</p>
|
3,781,193 | <p>For example, if we divide 100 by 50, then 100 by 49.8, then 49.8, etc. down to 100 divided by 1, we will have a list of 491 quotients, 10 of which are integers (2, 4, 5, 8, 10, 20, 25, 40, 50, 100). For the first 250 divisors (50.0 through 25.1), there is only one integer quotient (2). For the last 41 divisors (5.0 ... | Carl Schildkraut | 253,966 | <p>Take a real number <span class="math-container">$r$</span>. You're asking how likely it is that, for a given integer <span class="math-container">$n$</span>, <span class="math-container">$n/r$</span> is within some tolerance <span class="math-container">$\epsilon$</span> (in your question, it's <span class="math-con... |
415,597 | <p>I am trying to understand the concept of a natural transformation by considering the following example, an exercise from Mac Lane's <a href="http://books.google.com/books/about/Categories_for_the_Working_Mathematician.html?id=cUNdcgAACAAJ" rel="nofollow"><em>Categories for the working mathematician</em></a> (p. 18, ... | Stephen | 146,439 | <p>As you write, you should define $T(f)(h)=f \circ h$ for a function $f: A \rightarrow B$ and $h \in T(A)=A^S$. Then $T(f \circ g)(h)=(f \circ g) \circ h=f \circ (g \circ h)=T(f)(T(g)(h))$. Also, $T(1)=1$ (where $1$ is the identity on $A$). </p>
<p>The other functor is simply the identity $1$, and you are to check th... |
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | wythagoras | 236,048 | <p>Vote here! [<strong>no downvotes</strong> please. Only upvotes count]</p>
<p>$$\Huge{\mathrm{No.}}$$</p>
|
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | Nobody | 17,111 | <p>In response to the OP's and @Zach466920's answers.</p>
<p>I am glad to hear that the OP found a solution. I went to the OP's site. I found it's fine. It may not be perfect, but usable. There can be some improvements, though. In theory, you can do it in Chinese if Stack exchange sites can do it in English. Congratul... |
2,004,935 | <blockquote>
<p>The straight line $y = m(x – a)$ will meet the parabola $y^2 = 4ax$
at two distinct real points for which values of $m$?</p>
</blockquote>
<p>The answer is given as $m \in \mathbb R - {\{0\}}$.</p>
<p>I tried to solve by using the method of discriminants as follows:</p>
<p>$\{m(x-a)\}^2=4ax$, an... | mvw | 86,776 | <p>$$
y^2=4ax
$$
So we just need $a \ne 0$ to have a parabola. Otherwise there would be no possibility to have the intersection with a line be two distinct points.</p>
<p>For the intersection we got the system
$$
y = m (x - a) \\
y^2 = 4ax
$$
we note $m \ne 0$ is needed for two intersection points, otherwise we would ... |
1,503,957 | <p>Okay, maybe I am just really bad with exponents or forgot how exponents work but how do you do these 2 problems, here's what I got so far. I need to state whether thee sequence is increasing, decreasing, and use the ratio rule and difference rule to figure it out. </p>
<blockquote>
<p><strong>Ratio rule</strong>:... | zhw. | 228,045 | <p>Sketch: Let $a,b\in \mathbb {R}^n.$ It's enough to show $f(a+tb)$ is a convex function of $t\in \mathbb {R}.$ Differentiate this twice with respect to $t$ to see this amounts to showing</p>
<p>$$(\sum_{k=1}^{n}e^{a_k + tb_k})(\sum_{k=1}^{n}b_k^2e^{a_k + tb_k}) \ge (\sum_{k=1}^{n}b_ke^{a_k + tb_k})^2.$$</p>
<p>This... |
696,859 | <p>Is there a closed form for $k$ in the expression
$$am^k + bn^k = c$$
where $a, b, c, m, n$ are fixed real numbers?</p>
<p>If there is no closed form, what other ways are there of finding $k$?</p>
<p>Motivation: It came up when trying to apply an entropy model to allele distribution in genetics. The initial populat... | qwr | 122,489 | <p>I believe a closed form would only be possible if you could express $n^k$ in terms of $m^k$. Otherwise you would need to find the roots of $am^k + bn^k -c$, which I don't think is possible with elementary functions.</p>
<p>An well-known way to approximate roots is with Newton's Method.</p>
|
975,076 | <p>I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P C$. Is there a good way to solve this? Is this even guaranteed to have a solution?</p>
<p>The only way I can think ... | Grigory Ilizirov | 184,328 | <h2>The solution in other words</h2>
<p>$C$ is rigid transformations therefore: $C^{-1}=C^T$
and</p>
<p>\begin{equation}
(1) T=C^TPC
\end{equation}</p>
<p>Use orthogonal decomposition for T and P:<br/> (According to (1) $T$ and $P$ have the same eigenvalues so diagonal matrix will be the same)</p>
<p>\begin{equati... |
931,851 | <p>In trying to prove that every tree, <em>T</em>, has at most one perfect matching, I came across this idea:</p>
<blockquote>
<p>Since the matchings are perfect, each vertex has degree $0$ or $2$ in the symmetric difference, so every component is an isolated vertex or a cycle.</p>
</blockquote>
<p>Why is this true... | Yuval Filmus | 1,277 | <p>First, why the claim is true. Take any vertex. If it's matches to the same vertex in both perfect matching, then it had degree zero on the symmetric difference. Otherwise it has degree two. </p>
<p>Second, after removing all isolated vertices in the symmetric difference, all vertices have degree two. Take any such ... |
215,752 | <p>Let $A$ (or $X$) be </p>
<p>$\log A \sim N(\mu,\sigma^2)$, (lognormal distribution) </p>
<p>I have to show</p>
<p>$$E[A] = \exp[\mu + (\sigma^2/2)]\mbox{ and }E[A^2] = \exp[2\mu + 2\sigma^2].$$ </p>
<p>Do I have to use mgf of the normal dist. ?</p>
<p>It is easy to show E[$A^2$] since it is the second order d... | Ken Dunn | 42,937 | <p>Yes, the mgf would be an easy way to do this. Just replace $t = 1$ and you get the mean.</p>
|
1,920,125 | <p>I just came to a realization that my entire view of how functions work might not be completely sound. I'll explain:</p>
<p>Coming from more of a CS background, I see functions $f(x)$ as an example, as taking some input, "storing" it in $x$, substituting this new value wherever $x$ comes up, and produces one output.... | 3x89g2 | 90,914 | <p>Since you come from a computer science background, probably this explanation makes more sense to you. Consider this pseudo-code:</p>
<p><code>
def f(x:Real_Number): Real_Number {
return f(x)
}
</code>
This describes your $f(x)$.</p>
<p><code>
def g(x:Function): Function {
return derivative(f)
}
</code></p>
<... |
338,090 | <p>A is a $100 \times 100$ matrix.</p>
<p>The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$</p>
<p>Find the rank</p>
| Rustyn | 53,783 | <p>The rank of the matrix $A$ is $2$. I've found it. I used the following commands in MATLAB: </p>
<blockquote>
<p>for i=1:100; <br>
for j=1:100;<br>
A(i,j) = i^2 + j^2; <br>
end<br>
end<br>
rank(A)</p>
</blockquote>
|
338,090 | <p>A is a $100 \times 100$ matrix.</p>
<p>The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$</p>
<p>Find the rank</p>
| Abhra Abir Kundu | 48,639 | <p>I think the rank is $2$.</p>
<hr>
<p>Fact *: Rank of a matrix = Column rank.</p>
<p>Fact 1:When we perform any row operation or column operation to a matrix then the rank of the matrix does not change.</p>
<p>Now we perform the column operations: Subtract the first column from all the columns.</p>
<p>Then the m... |
2,809,519 | <p>Question: Prove that $\pi$ is irrational, assuming the following result: if $x$ is rational, $tan(x)$ is not. </p>
<p>Proof: Let $x$ $\in$ $\mathbb Q$ </p>
<p>I have seen Lambert's proof, however I am severely confused on where to begin. Do I now suppose $tan(x)$ is rational, then proceed by contradiction? Hints ... | hamam_Abdallah | 369,188 | <p>$$\tan (\pi)=0\in \mathbb Q $$</p>
<p>$$\implies \pi \notin \mathbb Q $$</p>
|
2,080,644 | <p>I know that a function is odd when
$$f(-x) = -f(x)$$
Therefore I can say that if for a function $$-f(x) + f(x) = f(-x) + f(x) = 0$$</p>
<p>Then the function is odd!</p>
<p>I tried to use this <em>trick</em> to prove that $f(x) = \ln\left(x+\sqrt{x^2 + 4}\right) - \ln2$ is odd.</p>
<p>However, I would want to prov... | ajotatxe | 132,456 | <p><strong>Hint:</strong></p>
<p>$$\frac{2}{\sqrt{x^2+4}-x}=\frac{2(\sqrt{x^2+4}+x)}4$$</p>
|
4,048,102 | <p>I came up with the following ODE:</p>
<p><span class="math-container">$\frac{dx}{dt}+\frac{dy}{dt}=k_1(a-x-y)(b-x)+k_2(a-x-y)(c-y)$</span></p>
<p>where <span class="math-container">$k_1,k_2,a,b,c$</span> are constants. I know under changes of variable, the equation can be changed into the following form:</p>
<p><spa... | fleablood | 280,126 | <p>You aren't proving: For every <span class="math-container">$n\in \mathbb N$</span>; we have <span class="math-container">$n \ge 4 \implies n!\ge 2^n$</span>.</p>
<p>You are proving: For every <span class="math-container">$n \ge 4; n \in \mathbb N$</span>; we have <span class="math-container">$n!\ge 2^n$</span>.</p>... |
4,421,316 | <p>Consider the sum,
<span class="math-container">$$\sum_{k=0}^n F_{4k}$$</span>
I would like to find this sum,
<span class="math-container">$F_n$</span> being the <span class="math-container">$n$</span>-th Fibonacci number.</p>
| user2661923 | 464,411 | <p>I am assuming that (in effect) you have <span class="math-container">$8$</span> pairs of socks, and each pair is a different color. So, you are in effect asking what the probability is that on all <span class="math-container">$3$</span> days, the two socks selected (<strong>without replacement</strong>) are differe... |
3,020,674 | <p>I need to use Fermat's Theorem to prove that 10001 is not prime. I understand that I just need to find a counterexample where <span class="math-container">$a^{10000}$</span> mod 10001 = 1 mod 10001 does not hold true, but this seems kind of difficult with such large numbers. Any ideas?</p>
| Barry Cipra | 86,747 | <p>This is a bit of a cheat, but another theorem of Fermat's says that <span class="math-container">$10001$</span> cannot be prime because it has <em>two</em> different representations as a sum of two squares:</p>
<p><span class="math-container">$$10001=100^2+1^2=65^2+76^2$$</span></p>
<p>The "cheat" here is that, wh... |
560,234 | <p>Given the series</p>
<blockquote>
<p>$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$</p>
</blockquote>
<p>I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$ might not exist, so the series is ... | AlmostSureUser | 103,374 | <p>I guess the standard argument should work. If $S_n=\sum_{k=0}^na_k$ converges then $a_k\rightarrow 0$. The necessary condition is not satisfied. </p>
|
3,744,801 | <p>I've been studying physics and I found this weird differentiation.</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\ln x = \ln a + \ln b$</span></p>
</blockquote>
</blockquote>
<blockquote>
<p>Now differentiating both sides,</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\dfrac{dx}x ... | Sameer Baheti | 567,070 | <p>You didn't apply the Leibniz Integral formula correctly.
<span class="math-container">$$\lim_{x\to0}\sin(e^x)+2\sin(e^{-2x})=\sin(e^0)+2\sin(e^0)=3\sin1$$</span></p>
|
3,744,801 | <p>I've been studying physics and I found this weird differentiation.</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\ln x = \ln a + \ln b$</span></p>
</blockquote>
</blockquote>
<blockquote>
<p>Now differentiating both sides,</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\dfrac{dx}x ... | user-492177 | 492,177 | <p>We have <span class="math-container">$\lim_{x\to 0} e^x=1$</span></p>
<p>Let let <span class="math-container">$\epsilon\gt 0$</span> be such that <span class="math-container">$0\lt 1-\epsilon\lt 1+\epsilon \lt 3.14\simeq \pi$</span></p>
<p>Notice that <span class="math-container">$\sin x$</span> is positive on <spa... |
3,744,801 | <p>I've been studying physics and I found this weird differentiation.</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\ln x = \ln a + \ln b$</span></p>
</blockquote>
</blockquote>
<blockquote>
<p>Now differentiating both sides,</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\dfrac{dx}x ... | Angina Seng | 436,618 | <p>If <span class="math-container">$F(x)$</span> is an antiderivative of <span class="math-container">$\sin(e^x)$</span>, that is <span class="math-container">$F'(x)=\sin(e^x)$</span>, then
<span class="math-container">$$I(x)=\int_{-2x}^x\sin(e^t)\,dt=F(x)-F(-2x).$$</span>
As <span class="math-container">$F(x)-F(-2x)\t... |
3,692,850 | <p>I want to know the convergence radius of <span class="math-container">$\displaystyle{\sum_{n=0}^{\infty}}(\sqrt{ 4^n +3^n}＋(-1)^n\sqrt{ 4^n-3^n})x^n$</span>.</p>
<p>Firstly, I tired to calculate <span class="math-container">$\lim_{k\to\infty}\left|\frac{a_{k}}{a_{k+1}}\right|$</span>,but I noticed this series does ... | Will Jagy | 10,400 | <p><span class="math-container">$${\sum_{n=0}^{\infty}} \left(\sqrt{ \; \; 1 \; \; + \left( \frac{3}{4} \right)^n \; \; \; } \; \;＋ \;(-1)^n \;\sqrt{ \; \; 1 \; \; - \left( \frac{3}{4} \right)^n \; \; \; } \; \;\right)(2x)^n$$</span>.</p>
|
3,692,850 | <p>I want to know the convergence radius of <span class="math-container">$\displaystyle{\sum_{n=0}^{\infty}}(\sqrt{ 4^n +3^n}＋(-1)^n\sqrt{ 4^n-3^n})x^n$</span>.</p>
<p>Firstly, I tired to calculate <span class="math-container">$\lim_{k\to\infty}\left|\frac{a_{k}}{a_{k+1}}\right|$</span>,but I noticed this series does ... | EDX | 763,728 | <p>Note that your principal terms are positive.</p>
<p><span class="math-container">$$ a_n = 4^{n/2} x^n(1+\dfrac{1}{2} \dfrac{3^n}{4^n}-\dfrac{1}{8}\dfrac{9^n}{16^n}+(-1)^n(1-\dfrac{1}{2} \dfrac{3^n}{4^n}+\dfrac{1}{8}\dfrac{9^n}{16^n}) + o(\dfrac{x^n9^n}{16^n})) $$</span></p>
<p><span class="math-container">$$a_{2n}... |
201,876 | <p>What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? Illustration from Wikipedia <img src="https://i.stack.imgur.com/oWM54.png" alt="enter image description here"> </p>
<p... | Vladimir Dotsenko | 1,306 | <p>I personally like this note of John Baez: <a href="http://math.ucr.edu/home/baez/braids/node4.html" rel="noreferrer">http://math.ucr.edu/home/baez/braids/node4.html</a></p>
|
2,282,578 | <p>Let $X$ have distribution function</p>
<p>$F_{X}(x)$ = \begin{cases}
0, & \text{if } a = 1, \\
\frac{1}{2}z^2, & \text{if 0 ≤ x ≤ 2}, \\
1, & \text{if } x > 2.
\end{cases}</p>
<p>Let $Y = X^2$. </p>
<p>Find $P(X + Y ≤\frac{3}{4})$.</p>
<p>All I have done is:</p>
<p>$P(X + Y ≤\frac{3}{4}) ... | Charbel | 504,828 | <p>I was wondering the same thing, why we don't apply what we learned in calculus, just set the partial derivatives to zero and find the minimal.</p>
<p>I think the main difference is:</p>
<ul>
<li>Partial derivatives set to zero will actually find the minima for your training examples. But once you found your parame... |
209,757 | <p>Find the value of $c$ which makes it possible to solve:</p>
<p>$$u+v+2w=2,$$
$$2u+3v-w=5,$$
$$3u+4v+w=c$$</p>
| Brian M. Scott | 12,042 | <p>Set up your augmented matrix in the usual way:</p>
<p>$$\left[\begin{array}{rrr|r}
1&1&2&2\\
2&3&-1&5\\
3&4&1&c
\end{array}\right]\;.$$</p>
<p>Then row-reduce it; reducing the first column, for instance, yields</p>
<p>$$\left[\begin{array}{rrr|c}
1&1&2&2\\
0&1&a... |
1,815,975 | <p>Someone is planning a round-the-world trip that involves visiting $2n$
cities, with two cities from each of $n$ different countries. He can choose a city to start and end the journey in, with the other $2n-1$ cities being visited exactly once. However, he has the restriction that the two cities from each c... | joriki | 6,622 | <p>Your edited version is correct but unnecessarily complicated. Consider the cities arranged in a circle. You want no two adjacent cities to be in the same country, and you want to distinguish one city as the start and end of the journey. There are $\binom nk$ ways to choose $k$ countries for which the constraint is v... |
666,103 | <p>I am studying elementary number theory, and just started learning about divisors. I always, try to read several other sources mostly because it helps me understand ideas better, also the textbook I am using- is not always clear for me. </p>
<p>Some sources state that 0|0 is not possible, while others allow 0|0. <... | egreg | 62,967 | <p>I see no reason whatsoever for excluding $0\mid 0$. The “divisibility” relation on the natural numbers
$$
a\mid b \quad\textit{for}\quad \text{there exists $c$ such that $b=ac$}
$$
is an order relation. I find the similarity with
$$
a\le b \quad\textit{for}\quad \text{there exists $c$ such that $b=a+c$}
$$
very appe... |
2,633,246 | <p>I need to find how many ways you can arrange the characters in Permutation with the $N$ appearing before all of the vowels but after the $P$. I understand that you can place the $N$ and $P$ $C(6,2)$ ways, but I'm not sure how to accommodate for the vowels. I could place the $N$, the $P$, and the vowels $(6,3)$ ways ... | Mayank Mittal | 349,312 | <p>Firstly forget one of T to make things easy as we can handle that later. Now consider 2 groups. The group after N and the group before N. Find the number of permutations of both the groups by shifting N towards left each time (with the first case as all vowels only in the forst group). Now we just need to combine th... |
1,525,470 | <p>Let $K$ be a finite separable extension of a field $k$ of prime degree $p$. Let $\theta$ in K be such that $K = k(\theta)$ and $\theta_1 ..., \theta_p$ be the conjugates of $\theta$ over $k$ in some algebraic closure $k^a$. Let $\theta=\theta_1$ and if $\theta_2 \in k(\theta)$ Then show that $K/k$ Galois.</p>
<p>As... | dragoboy | 168,033 | <p>Let $|Aut(k(\theta):k|=a$. Consider $E=k(\theta_1,\theta_2,..,\theta_p)$. Clearly $E/k$ is Galois. Now as $E/k$ is separable, so we can let $[E:k(\theta)]_{s}=[E:k(\theta)]=n$. </p>
<p>Now consider the set $S \subset G$ such that $S=\{\sigma \in G \mid \sigma_{k(\theta)}\in Aut(k(\theta)/k)\}$. Clearly, $S$ is sub... |
169,998 | <p>If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within the angle created by those points?</p>
<p>The process I'm currently using is to get the angle of all three lines tha... | Ross Millikan | 1,827 | <p>The two lines divide the plane into four wedges. The fourth point will be in one of these four wedges. To find which one, you can use the <a href="http://en.wikipedia.org/wiki/Atan2" rel="nofollow">Atan2</a> function to find the angle of each line and the angle of the fourth point from the origin. You can look to ... |
1,973,500 | <p>What is the number of abelian groups of order 40? I thought the number is just $3$. More specifically, they are</p>
<p>$$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_5$$
$$\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_5$$
$$\mathbb{Z}_8\times\mathbb{Z}_5$$</p>
<p>However, my answer says:<a... | Asinomás | 33,907 | <p>Your book is wrong, $\mathbb Z_{10}$ is congruent to $\mathbb Z_5\times \mathbb Z_2$.</p>
<p>So $\mathbb Z_{10}\times \mathbb Z_2\times \mathbb Z_2$ is congruent to $\mathbb Z_5\times \mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$</p>
|
267,618 | <p>So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$?
Using the Chernoff bound we know that $\frac{1}{\epsilon^2}\log(1/\delta)$ is enough. And I know that this is als... | Aryeh Kontorovich | 12,518 | <p>In this paper with Iosif Pinelis, we give the exact constants for the lower bound (whose dependence on $\epsilon$ is $\frac1{\epsilon^2}$):
<a href="https://arxiv.org/abs/1606.08920" rel="nofollow noreferrer">https://arxiv.org/abs/1606.08920</a></p>
<p>Actually, our result is stronger since the bound holds for <em>... |
3,784,663 | <p>Pretty self explanatory.</p>
<blockquote>
<p>If I had for example something like <span class="math-container">$\sin^{-1}(\pi/12)$</span> in an expression, is it ever possible to express that expression without inverse trig functions?</p>
</blockquote>
| IV_ | 292,527 | <p>You can give your number a name or represent it explicitly e.g. as value of another function or of an integral, as limit of a series, or implicitly as solution of an equation.<br />
<span class="math-container">$\ $</span></p>
<p>The numbers that are explicitly representable by finite terms are the explicit ones amo... |
769,678 | <p>While I'm studying set theory, I saw the definition of indexed set product in <a href="http://en.wikipedia.org/wiki/Cartesian_product#Infinite_products" rel="nofollow">Wikipedia</a>.</p>
<p>The definition looks complicated to me, so I tried to define simpler. The definition is below.</p>
<p>$$\prod_{i\in\mathcal{I... | Martin Argerami | 22,857 | <p>It looks more complicated to me. Your unions are precisely the functions $\mathcal I\to\bigcup X_i$, just written in a less intuitive way. When you write $\{x_i\}_{\mathcal I} $, that's nothing but a function, which you would see more clearly if you wrote $x(i) $ instead of $x_i$ (I'm not suggesting you actually do... |
769,678 | <p>While I'm studying set theory, I saw the definition of indexed set product in <a href="http://en.wikipedia.org/wiki/Cartesian_product#Infinite_products" rel="nofollow">Wikipedia</a>.</p>
<p>The definition looks complicated to me, so I tried to define simpler. The definition is below.</p>
<p>$$\prod_{i\in\mathcal{I... | drhab | 75,923 | <p>Let $I$ be a set and let $X_{i}$ be a set for each $i\in I$. </p>
<p>What can serve as <em>product</em> for these sets is a <em>set</em> $X$ <em>together with functions</em>
(<em>projections</em>) $p_{i}:X\rightarrow X_{i}$ for $i\in I$ in
such a way that there is a one-to-one correspondence between so-called
'sour... |
3,541,869 | <p>I was reading from <a href="https://books.google.com.gh/books/about/Ordinary_Differential_Equations.html?id=iU4zDAAAQBAJ&source=kp_book_description&redir_esc=y" rel="nofollow noreferrer"><em>Ordinary Differential Equations</em></a> <strong>(Lesson 13 Example 13.3 page 110)</strong> and came across this quest... | ms_ | 615,427 | <p>I'll go ahead and reproduce the proof with arguments to answer your questions.</p>
<p>Assuming your assumptions, we get that for <span class="math-container">$x \in A$</span>
<span class="math-container">$$
||(f+g)(x)-(b+c)||=||f(x)+g(x)-(b+c)||=||f(x)-b+g(x)-c)||
$$</span>
<span class="math-container">$$
\leq||f(x... |
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| pre-kidney | 34,662 | <p>Another way to phrase this:
The product ideal $IJ$ is the <em>smallest</em> ideal containing all the products of elements of $I$ with elements of $J$.</p>
<p>As for examples: In $\mathbb{Z}$, we have $$\langle a\rangle\langle b\rangle=\langle ab\rangle$$</p>
|
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| Jonathan | 37,832 | <p>One would like the product ideal to be
$$IJ=\{ij\mid i\in I,j\in J\}$$
but we can easily see that there is a problem. It must be closed under addition, so $ij+i'j'$ must be in $IJ$. Can you find $i''\in I$, $j''\in J$ such that $ij+i'j'=i''j''$ so that it's in $IJ$ as defined above? Not in general, no. The natural w... |
2,087,596 | <p>Find the discontinuity at $f(2)$ of the function $f(x)=\dfrac{x^2-3x+2}{{x^2}+x-6}$.</p>
<p>I am confused. I do not understand that is there discontinuity at $2$ but it has discontinuity at $x=-2$. can you explain it please? For my point of view, there is no discontinuity at $2$ because after factorization I get $f... | Simply Beautiful Art | 272,831 | <p>By definition, the anti-log is the inverse of the log:</p>
<p>$$\operatorname{antilog}_a(y)=x$$</p>
<p>Take the log of both sides:</p>
<p>$$\log_a(\operatorname{antilog}_a(y))=\log_a(x)$$</p>
<p>They cancel, and we are left with</p>
<p>$$y=\log_a(x)$$</p>
|
2,087,596 | <p>Find the discontinuity at $f(2)$ of the function $f(x)=\dfrac{x^2-3x+2}{{x^2}+x-6}$.</p>
<p>I am confused. I do not understand that is there discontinuity at $2$ but it has discontinuity at $x=-2$. can you explain it please? For my point of view, there is no discontinuity at $2$ because after factorization I get $f... | Raknos13 | 396,929 | <p>If $$antilog_a(y) = x $$
$$\Rightarrow a^{y} = x $$</p>
<p><strong>credit</strong>: <em>John</em> & <em>J. M</em> from the comments </p>
|
3,314,667 | <h1>Problem</h1>
<p>Given sets <span class="math-container">$\mathcal A$</span>, <span class="math-container">$\mathcal B$</span>, <span class="math-container">$\mathcal Y$</span>, let <span class="math-container">$\mathcal X$</span> be a set with the following properties:</p>
<ul>
<li><span class="math-container">$\... | fleablood | 280,126 | <p>How could that possibly work? </p>
<p>If <span class="math-container">$a,b \in G \subset H$</span> and <span class="math-container">$H$</span> is abelian then <span class="math-container">$a*b =b*a$</span> so <span class="math-container">$G$</span> is abelian.</p>
<p>The "tainting" issue should be obvious. If <s... |
9,022 | <p>Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to figure out why. Is it obvious? A simple yes or no (with reference is possible) is all I need.</p>
| user717 | 717 | <p>In fact, an even stronger statement holds: If $G$ is a topological group and $H$ is an (abstract) subgroup, then $G/H$ is Hausdorff if and only if $H$ is closed (cf Bourbaki, General Topology, III.2.5, prop 13). It's not hard to prove.</p>
|
4,550,913 | <p>I want to show that <span class="math-container">$f:[0,2\pi]\longrightarrow\mathbf{S}^1$</span> defined as <span class="math-container">$f(x)=(\cos x,\sin x)$</span> is closed, surjective, and continuous but not open. I already prooved that it is surjective but I still can't show that it is closed and not open.</p>
| Sam | 530,289 | <p>The set <span class="math-container">$[0,\pi)$</span> is open in <span class="math-container">$[0,2\pi]$</span> since it can be written as <span class="math-container">$(-\pi,\pi)\cap[0,2\pi)$</span></p>
<p>On the other hand, <span class="math-container">$f([0,pi))$</span> is the upper half of the circle including <... |
1,493,874 | <p>How can solve that logarithms</p>
<p>$\log _{\frac{4}{x}}\left(x^2-6\right)=2$</p>
<p>It's look diffucult to solve </p>
<p>I was solve but stop with</p>
<p>$x^4−6x^2−16=0$</p>
<p>what is next?</p>
| E.H.E | 187,799 | <p>Hint:
$$(\frac{4}{x})^2=(x^2-6)$$
$$16=x^4-6x^2$$
$$x^4-6x^2-16=0$$
then solve it by using the quadratic formula
$$x^2=3\pm\sqrt{9+16}$$
$$x^2=3\pm5$$
$$x^2=8$$
$$x=\pm2\sqrt{2}$$
or
$$x^2=-2$$
$$x=\pm\sqrt{2}i$$</p>
|
1,554,466 | <blockquote>
<p>$$Log(-e i)$$</p>
</blockquote>
<p><strong>My try:</strong></p>
<p>$$=\ln|0+(-e)i|+i[\arg (0+(-e i))+2\pi k]$$</p>
<p>$$=\ln|e i|+i(-\frac{\pi}{2}+2\pi k)$$</p>
<p>My attempt is correct?</p>
| Justpassingby | 293,332 | <p>Correct but incomplete. The absolute value of ei is e, and the ln of that is 1.</p>
|
1,554,466 | <blockquote>
<p>$$Log(-e i)$$</p>
</blockquote>
<p><strong>My try:</strong></p>
<p>$$=\ln|0+(-e)i|+i[\arg (0+(-e i))+2\pi k]$$</p>
<p>$$=\ln|e i|+i(-\frac{\pi}{2}+2\pi k)$$</p>
<p>My attempt is correct?</p>
| 3SAT | 203,577 | <p>$$...=\ln(|i||e|)$$</p>
<p>$$=\ln(|i|)+\ln(|e|)$$</p>
<p>$$=\ln(1)+\ln(|e|)$$</p>
<p>$$=0+1$$</p>
<p>$$\Longrightarrow \boxed{1+i(-\frac{\pi}{2}+2 \pi k)}$$</p>
|
1,089,593 | <p>How to solve $\dfrac{dy}{dx}=\cos(x-y)$ ? How do I separate x and y here ?</p>
<p>Please advise.</p>
| Mike | 17,976 | <p>A substitution might be best.</p>
<p>$$z=x-y,z'=1-y'$$</p>
<p>$$1-z'=\cos z,z'=1-\cos z$$</p>
<p>$$x=\int\frac{dz}{1-\cos z}$$</p>
<p>From here, probably multiply top and bottom by $1+\cos z$.</p>
|
4,222,999 | <p>I want to know a general way to express the PDF of <span class="math-container">$Y = X^2$</span> for <span class="math-container">$X\sim U(a,b)$</span> arbitrary <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that <span class="math-container">$a < b$</span> and <span clas... | Graham Kemp | 135,106 | <p>Because <span class="math-container">$X\mapsto X^2$</span> has two semi-inverses, the Jacobian transformation is:<span class="math-container">$$\begin{align}f_{X^2}(y)&=\left|\frac{d(\surd y)}{dy}\right|f_{X}(\surd y)+\left|\frac{d(-\surd y)}{dy}\right|f_{X}(-\surd y)\\&=\frac{\mathbf 1_{a\leq\surd y\leq b}+... |
4,075,530 | <p>Let <span class="math-container">$S$</span> be a closed orientable surface. For any positive integer <span class="math-container">$n$</span>, is there a connected <span class="math-container">$n$</span>-sheeted covering space of <span class="math-container">$S$</span>? This is certainly not true if <span class="math... | Yuval | 609,959 | <p>Yes. See the picture for an example that a surface of genus 3 got a 5-sheeted covering:<a href="https://i.stack.imgur.com/4JQ1h.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4JQ1h.png" alt="enter image description here" /></a></p>
<p>This lay out works for any surfaces of genus <span class="math... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Mark | 24,958 | <p>From Euler's <span class="math-container">$$\sin[2x]=\Im[e^{xi+xi}]=\Im[[\cos x+i\sin x][\cos x+i\sin x]]=2\sin[x]\cos[x]$$</span>
<span class="math-container">$$\frac{\sin[πx]}{πx}
=\prod_{n=1}^\infty\left[1^2-\left[\frac{x}n\right]^2\right]$$</span>
we get <span class="math-container">$$\cos\left[\frac\pi2x\right]... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Nikolaj-K | 18,993 | <p>OP here. I just realized that the following should hold in general:</p>
<p>$$\lim_{n\to\infty}a_n=a_1+\sum_{n=1}^\infty(a_{n+1}-{a_n}),$$</p>
<p>and for finite $a_n$, similarly</p>
<p>$$\lim_{n\to\infty}a_n=a_1\cdot\prod_{n=1}^\infty\frac{a_{n+1}}{a_n}.$$</p>
<p>Hence, with $a_1=\prod_{n=1}^\infty (a_1)^{2^{-n}}... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Community | -1 | <p>Amazingly, the exponential function can be represented as an infinite product of a product! That result was shown in the 2006 paper "Double Integrals and Infinite Products For Some Classical Constants Via Analytic Continuations of Lerch's Transendent" by Jesus Guillera and Jonathan Sondow.</p>
<p>It is proven in Th... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | John Bentin | 875 | <p>We have<span class="math-container">$$\mathrm e^{-x}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$</span><span class="math-container">$$(1-x)(1+\tfrac12x^2)(1+\tfrac13x^3)(1+\tfrac38x^4)(1+\tfrac15x^5)(1+\tfrac{13}{72}x^6)(1+\tfrac17x^7)(1+\tfrac{27}{128}x^8... |
1,584,653 | <p>Let linear transformation is defined as </p>
<p>$\mathcal{A}(1,1,1)=(1,0,0)$</p>
<p>$\mathcal{A}(1,-1,0)=(1,1,1)$</p>
<p>$\mathcal{A}(1,0,1)=(1,1,1)$</p>
<p>Find matrix of $\mathcal{A}$ and inverse (not in matrix representation, if exists).</p>
<p>Attempt:</p>
<p>Transformation $\mathcal{A}$ can be represented... | Jeb | 136,806 | <p>Hint: Notice that if you call
$$ e_1 = \begin{pmatrix} 1 \\ 1 \\1 \end{pmatrix} , \quad e_2 = \begin{pmatrix} 1 \\ -1 \\0 \end{pmatrix} , \quad e_3 = \begin{pmatrix} 1 \\ 0 \\1 \end{pmatrix} $$
We have that
$$ e_1 + e_2 - e_3 = \begin{pmatrix} 1 \\ 0 \\0 \end{pmatrix} $$
$$ e_1 - e_3 = \begin{pmatrix} 0 \\ 1 \... |
1,487,878 | <blockquote>
<p>Prove: $\sqrt[4]{4}$ is irrational</p>
</blockquote>
<p>I know that $\sqrt{p}$ is irrational where $p$ is a prime number.</p>
<p>So $\sqrt[4]{4}=\sqrt[4]{2*2}=16*\sqrt[4]{2}=16*2^{\frac{1}{2}^\frac{1}{2}}$ </p>
<p>What can I say about $2^{\frac{1}{2}^\frac{1}{2}}$?</p>
| Renato Faraone | 217,700 | <p>There are a few misunderstandings here:</p>
<p>1)The main issue is that $\sqrt[4]4=\sqrt[4]2\times\sqrt[4]2\neq 16\times\sqrt[4]2$ because that would imply that $\sqrt[4]2=16$ (you have confused exponentiation with roots).</p>
<p>2)While it is true that $\sqrt[4]2=(2^{\frac 12})^{\frac 12}$ it isn't that $\sqrt[4]... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Will Jagy | 10,400 | <p>I have been asked for a hint. Gauss was thinking about roots of unity. The method of Gauss for dealing with these polynomials is in chapter 9 of <a href="http://oskicat.berkeley.edu/record=b18634773~S1" rel="nofollow noreferrer">Cox Galois Theory</a>. In fact, as the author points out, this work predates Galois Theo... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Austin Mohr | 11,245 | <p><a href="https://projecteuler.net/" rel="noreferrer">Project Euler</a> is a great source of interesting problems. Many of them require you to learn a little computer programming, which I highly recommend you try if you haven't before. (And if you don't have a preferred programming language, give <a href="https://clo... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Argon | 27,624 | <p>These are some integral-themed questions I had fun with at your age.</p>
<p>Evaluate </p>
<p>$$\int_0^{\pi}\log\sin x\,dx$$</p>
<p>$$\int_0^{\pi/2}\frac{\sin^a(x)}{\sin^a(x)+\cos^a(x)} \,dx\\\vphantom{\cfrac11}$$</p>
<p>Show that:</p>
<p>for $n\in \mathbb Z^{\ge0}$</p>
<p>$$n! = \int_0^{\infty}x^ne^{-x} \,dx$$... |
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